一个特殊子簇与Hecke平移的不可能相交
Unlikely intersections with Hecke translates of a special subvariety
摘要：
我们证明了一些情况下的Zilber-Pink猜想不太可能的交叉点志村品种。首先，我们证明了Zilber-Pink猜想对于曲线与固定特殊子簇的Hecke平移的并之间的交成立，条件是算术代数。其次，我们证明了猜想无条件的交叉之间的曲线和工会的Hecke对应的模空间的主要极化阿贝尔簇，受到一些技术性的假设。这推广了Habegger和Pila关于模曲线乘积的Zilber-Pink猜想的结果。条件证明使用Pila-Zannier方法，依赖于Habegger和Pila的点计数定理和Gao的函数超越结果。无条件的结果推导出使用各种算术成分：的Masser-Wüstholz issues定理，比较之间的Faltings和Weil高度，超逼近定理的Salehi Golsefidy，和结果的扩展和gonality由于Ellenberg，霍尔和Kowalski。
英文摘要：
We prove some cases of the Zilber-Pink conjecture on unlikely intersections in Shimura varieties. Firstly, we prove that the Zilber-Pink conjecture holds for intersections between a curve and the union of the Hecke translates of a fixed special subvariety, conditional on arithmetic conjectures. Secondly, we prove the conjecture unconditionally for intersections between a curve and the union of Hecke correspondences on the moduli space of principally polarised abelian varieties, subject to some technical hypotheses. This generalises results of Habegger and Pila on the Zilber-Pink conjecture for products of modular curves. The conditional proof uses the Pila-Zannier method, relying on a point-counting theorem of Habegger and Pila and a functional transcendence result of Gao. The unconditional results are deduced from this using a variety of arithmetic ingredients: the Masser-Wüstholz isogeny theorem, comparison between Faltings and Weil heights, a super-approximation theorem of Salehi Golsefidy, and a result on expansion and gonality due to Ellenberg, Hall and Kowalski.
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